Uh, Craig? Now you have confused me.
If I put two clips, untrimmed, so no handles, on the timeline, and drop a Dip To Black transition on the cut, it takes it. It complains about it but it takes it - centered on the cut. And it works. It just complains. But it works. No "At Start" or "At End" needed. I moved the error message closer to the timeline to put it into one screen shot, but you can see that it is centered on the cut.
And when I play the timeline back, the clips do not repeat frames. I have to watch carefully in the darker frames but as far as I can tell, the motion is still smooth. (I changed videos since I decided to show you an example and the one I did the screen shot for had a child in it). I extended the Dip to Black so you could see that it got darker and darker and then lighter and lighter but no frames repeated.